The package SMC-Solver for solving structured Markov chains is presented. It contains the most advanced algorithms for solving QBD, M/G/1 and G/M/1 problems. The package is provided in two versions: a Matlab toolbox and a Fortran 95 version with a user-friendly graphical interface.

Objective: The traditional surgical approach for patients with primary hyperparathyroidism (PHPT) consists of the identification of at least four glands and in the removal of all hyperfunctioning parathyroid tissue. Design: To evaluate whether intraoperative parathyroid hormone (PTH) monitoring will allow a more limited surgical procedure by confirming complete removal of all hyperfunctioning tissue. Methods: Plasma samples were obtained from 206 consecutive patients with sporadic PHPT before skin incision, during manipulation of a suspected adenoma, and 5 min (T-5) and 10 min after removal of abnormal parathyroid tissue. PTH was measured by a quick immunochemiluminescent assay (QPTH). The operative success was defined by a decrease of PTH greater than 50% of the highest pre-excision value. Results: A .50% decrease of PTH occurred in 203 patients and was evident at T-5 in the majority of cases. All but three had normal serum calcium the day after surgery and afterwards. PTH concentration did not show a . 50% decrease in the remaining three cases after completion of surgery. One patients had negative neck exploration and remained hypercalcemic; the other two had normal serum calcium at follow-up. Thus, the intraoperative QPTH correctly predicted the outcome of surgery in 201 patients (97.5%) (200 true positive and 1 true negative), and provided three false positive and two false negative results. Conclusions: The intraoperative QPTH measurement represents a useful tool to assist the surgeon during parathyroidectomy. It indicates whether all hyperfunctioning parathyroid tissue has been removed, limiting the procedure to a unilateral neck exploration in most cases.

We analyze the problem of the numerical solution of structured Markov chains encountered in queuing models: we describe the main computational problems and present the most advanced algorithms currently available for their solutions.

We expand and update the software tool SMCSolver, presented at the SMCTools workshop in 2006, for the numerical solution of structured Markov chains encountered in queuing models. In particular the new version of the package implements different transformation techniques and different shift strategies which are combined in order to speed up and optimize the solution of structured Markov chains. Numerical experiments show the effectiveness of the new implemented techniques.

Matrix-analytic methods have advanced considerably since the pioneering work of Marcel Neuts [6, 5] on Quasi-Birth-Death (QBD), GI/M/1- and M/G/1- type Markov chains (MCs). Especially the algorithms involved to (iteratively) solve these structured Markov chains have matured a lot, which has resulted in more efficient, but also more complex algorithms [4, 1]. While the first algorithms were straightforward to implement---as they were based on simple functional iterations---more advanced algorithms/features like cyclic-reduction, the Newton iteration or the shift technique (to accelerate convergence), require more effort; in particular for GI/M/1- and M/G/1-type Markov chains. This has motivated us to develop the Structured Markov Chain Solver (SMCSolver) tool [2], which implements a large number of basic and more advanced algorithms for solving QBD, GI/M/1- and M/G/1-type MCs1 (as well as the more general Non-Skip-Free M/G/1-type MCs). The MATLAB version of the tool consists of a collection of MATLAB functions, while the Fortran version is accompanied by a graphical user-interface (GUI). Apart from making these more advanced algorithms accessible to non-specialists, the tool is also useful as a platform for the development and study of new algorithms and acceleration techniques. Since its initial release in 2006, various extensions have been made. In [3] different transformation techniques and shift strategies are incorporated in order to speed up and optimize the algorithms, while even more recently an efficient Newton iteration for GI/M/1- and M/G/1-type Markov chains was included [8]. Matrix-analytic methods have also been very effective in the analysis of many queueing systems in both discrete- and continuous-time. The Q-MAM tool [7] is a collection of MATLAB functions that allows one to compute the queue length, waiting time and delay distribution of various queueing systems of infinite size. It includes amongst others implementations of the PH/PH/1, MAP/MAP/1, MAP/M/c, MAP/D/c, RAP/RAP/1, MMAP[K]/PH[K]/1, MMAP[K]/SM[K]/1, SM[K]/PH[K]/1 (many in both discrete- and continuous-time), where state-of-the-art solution techniques are used to solve these models efficiently. The Matlab version of the SMCSolver and Q-MAM tool is available at http://win.ua.ac.be/?vanhoudt/ while the Fortran 90 version of the SMCSolver tool with the GUI can be downloaded from http://bezout.dm.unipi.it/SMCSolver.